Local-to-nonlocal transition laws: generalising mass-conservation to two-dimensional nonlocal lattices

Metamaterials, artificial lattices with uncommon dynamical properties, have received growing attention due to their wave manipulation capacities. In particular, engineering their dispersion curves allows for obtaining targeted, band-specific responses, with applications in a broad range of subjects. Roton-like dispersion relations, which feature local minima and maxima that invert the direction of energy propagation, have broadened the reach of these devices, with one way to achieve them being the inclusion of beyond-next-nearest neighbours. The mass-spring models usually employed to simulate metamaterial behaviour traditionally assume massless springs. However, in nonlocal cases, this hypothesis may not be always reasonable, and to ensure physical soundness, a mass conservation law for spring mass becomes crucial to prevent their total mass from significantly affecting system dynamics. With the aim of gaining insights into two-dimensional nonlocal systems, this work extends the previous model by the present authors from one-dimensional monoatomic chains to two-dimensional lattices. Building on the mass conservation principle proposed in the above-mentioned work, a mechanical consistency condition is applied to two-dimensional discrete periodic lattices. Two dimensionless parameters are introduced into the analysis: α, to adjust stiffness distribution, and the nonlocality level P, established through homothety. Analytical dispersion relations as functions of α and P are derived for the five Bravais lattices, and the effects of these parameters on dynamical behaviour are discussed. The potential for applications in waveguiding and lensing of two-dimensional metamaterials is undoubted, and through tuneable nonlocal models, new possibilities for advanced devices may be unlocked.